Differential graded algebras for trivalent plane graphs and their representations
Abstract
To any trivalent plane graph embedded in the sphere, Casals and Murphy associate a differential graded algebra (dg-algebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is the construction of a generalization of the Casals--Murphy dg-algebra to non-commutative coefficients, for which we prove various functoriality properties not previously verified in the commutative setting. Our second result is to prove that rank $r$ representations of this dg-algebra, over a field $\mathbb{F}$, correspond to colorings of the faces of the graph by elements of the Grassmannian $\operatorname{Gr}(r,2r;\mathbb{F})$ so that bordering faces are transverse, up to the natural action of $\operatorname{PGL}_{2r}(\mathbb{F})$. Underlying the combinatorics, the dg-algebra is a computation of the fully non-commutative Legendrian contact dg-algebra for Legendrian satellites of Legendrian 2-weaves, though we do not prove as such in this paper. The graph coloring problem verifies that for Legendrian 2-weaves, rank $r$ representations of the Legendrian contact dg-algebra correspond to constructible sheaves of microlocal rank $r$. This is the first explicit such computation of the bijection between the moduli spaces of representations and sheaves for an infinite family of Legendrian surfaces.
- Publication:
-
arXiv e-prints
- Pub Date:
- October 2021
- DOI:
- 10.48550/arXiv.2110.07585
- arXiv:
- arXiv:2110.07585
- Bibcode:
- 2021arXiv211007585S
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Geometric Topology;
- Mathematics - Symplectic Geometry;
- 53D10;
- 05C15;
- 53D42;
- 57R17
- E-Print:
- Added Section 1.4 on the underlying contact geometry. Section 2 streamlined. Lemma 3.2 fixed so that it applies as desired. Appendix included with a sample computation. Version accepted in Quantum Topology